Representing a function of one variable in terms of a sequence of gaussians

[thoughtgaze]

Is this possible? It seems like it should be but, it's difficult to find an explicit relationship between a general function of one variable x (let's say we are only interested in functions that decay to zero as they go to plus or minus infinity)

it seems like summing a bunch of gaussians of arbitrary width located at various points along the domain, one should be able to construct an arbitrary function but I don't see much useful material on it.

 

[Stephen Tashi]

Look at the topic of "radial basis functions" and think of the 1-dimensional case.

You also might find the topic of "wavelets" interesting. I don't know whether one can make useful "wavelets" with a gaussian shape.

 

[Xiuh]

Radial basis functions are the answer if by "representing" you mean "interpolating". Let $f$ be a function and consider the interpolant
$$\sigma(x) = \sum_{j=1}^N a_j e^{-\frac{(x-x_j)^2}{c^2} },$$
for some $c$ and some set of nodes $\{x_j\}$. The interpolation conditions $f(x_j) = \sigma(x_j)$ give you a system of equations for the coefficients $a_j$. This system always has a solution, because the gaussian is positive definite. However, note that this is not an exact representation, you are just interpolating.